A386279 Number of values between 0 and n-1 which never appear in the sequence {b(m), m>0}, defined by b(m) = binomial(m+n,n) mod n.

0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, 6, 0, 0, 8, 0, 12, 6, 10, 0, 8, 0, 12, 0, 16, 0, 0, 0, 0, 12, 16, 20, 0, 0, 18, 12, 0, 0, 30, 0, 24, 0, 22, 0, 0, 0, 24, 18, 32, 0, 24, 30, 0, 18, 28, 0, 28, 0, 30, 14, 0, 20, 42, 0, 40, 24, 28, 0, 32, 0, 36, 30, 44, 42, 58, 0, 0, 0, 40

Offset

1,6

Comments

The sequence {b(m), m>0} is always periodic (see A133900 and A133910), so one need only compute the first period to find a(n).

a(n) can be arbitrarily large.

a(n)/n can be arbitrarily close to 1.

The natural density of n such that a(n)>0, assuming it exists, is greater or equal to log(2).

a(n) is always even.

Links

Table of n, a(n) for n=1..82.

Formula

a(n) = 0 if n is prime or a power of a prime.

a(2p) = p-1 if p>2 is prime.

Example

For n = 3, the sequence {b(m), m>0} (see A133873) has all three residues, 0, 1, and 2, so a(3)=0.
For n = 6, the sequence {b(m), m>0} (see A133886) does not have the residues 2 and 5, so a(6)=2.

Prog

(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 2] += logint(n, f[k, 1])); my(nb=factorback(f), v=vector(nb, k, binomial(k+n, n) % n)); n - #Set(v); \\ Michel Marcus, Oct 13 2025

Crossrefs

Cf. A133873, A133875, A133877, A133884, A133886, A133888, A133889, A133890, A133900, A133910.

Sequence in context: A135416 A134309 A051516 * A236799 A208274 A208604

Adjacent sequences: A386276 A386277 A386278 * A386280 A386281 A386282

Keyword

nonn

Author

Hari Rajesh, Oct 12 2025

Status

approved